plot([coord(1:nelem) coord(2:nelem+1)]’,sigma’)
elseif norder==2,
plot([coord(1:2:2*nelem) coord(2:2:2*nelem+1)
coord(3:2:2*nelem+2)]’,sigma’)
end
xlabel(’x’), ylabel(’\sigma’);
(b)
ii=pos(2,:) ; u = sol(nonzeros(ii))
; u = [0.2 ; 0.4]
(c)
ii=dest(3,:) ; u=sol(ii) ; u=0.4
36
Answers to the exercises of chapter 14
14.8
(a)
d
dx
c
du
dx

f
=
0
u
(0)
=
5
u
(
l
)
=
5
(b)
u
(
x
) =
f
2
c
x
2

fl
2
c
x
+ 5
(c) When the thickness
t
= 5mm the glucose level in the middle
of the construct will become less than zero (in reality of course
zero) meaning that eventually cells will die in the middle of
the construct.
14.9
(a) Add:
xx=n(int,:)*nodcoord;
a1=1.6;
% in [cm]
a2=0.15;
% in [cm^1]
a3=0.8;
% dimensioneless
E=10;
% in [Ncm^2]
r=a1*(sin(a2*xx+a3))^3;
A=pi*r^2 %calculate surface of cross section
c=E*A;
% product of cross section and Youngs modulus
in the element routine.
(b) See figure
14.3
.
Exercises
37
0
2
4
6
8
10
12
0
1
2
3
4
5
6
7
x
u
Fig. 14.3. Exercise 14.9: Displacement as a function of x
15
Answers to the exercises of chapter 15
Exercises
15.1
(b) See Fig.
15.1
. For small
a
the solution is approximately the
same as the solution of the diffusion equation.
For larger
a
particles drag along with the fluid and thus the higher con
centrations shift to the right. For values higher than 10 the
solution becomes unstable.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
u
a = 0.001
(a)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
u
a = 1
(b)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.5
1
x
u
a = 10
(c)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.4
0.2
0
0.2
0.4
0.6
0.8
1
x
u
a = 20
(d)
Fig. 15.1.
Solution of the steady convection diffusion equation for different
values of
a
using 5 linear elements
38
Exercises
39
(c) With
h
= 0
.
2 and
c
= 1 it follows that
Pe
h
>
1 if
a >
10.
15.2
(a) It can be seen that using
θ
= 0
.
5 the solutions remain stable,
even for large time steps. When
θ <
0
.
5 the solution starts
to oscillate when the time step becomes to large.
(b) When
a >
20 solutions will become unstable.
(c) When Δ
t
= 0
.
001 the discretization of
u
no longer allows an
”exact” description of the
u
as a function of
x
.
(d) Reducing the convective velocity does not influence the sta
bility much. Increasing the number of elements (i.e. reducing
the size of an element) again leads to stable solutions.
0
0.2
0.4
0.6
0.8
1
1
0.5
0
0.5
1
x
u
Fig. 15.2. Solution for different time steps
15.3
(a) Initial conditions. Use sol(:,1) (second index is the time step)
For initial conditions the following script can be used:
for inode = 1: nnodes
idof=dest(inode,1)
x = coord(inode,1)
sol(idof,1)=sin(2*pi*x)
end
Essential boundary condition
u
= 0 at
x
= 0;
bndcon=[1 1 0]
40
Answers to the exercises of chapter 15
Natural boundary condition:
c ∂u/∂x
= 0 at
x
= 1. It is not
necessary to prescribe this, because default this condition is
set when no essential boundary conditions are prescribed.
(b) The same as in item (a) but with different initial conditions.
15.4
(a) For the initial condition add the lines:
ii=find(coord>0.0001) % excludes the point at x=0
sol(dest(ii,1))=1000;
This change can be made in the file:
fem1dcd
or in
demo_fem1dcd
.
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